A Bernstein-type estimator for decreasing density with application to p -value adjustments

• Published in: Computational statistics & data analysis Vol. 56; no. 2; pp. 427 - 437
• Main Authors: Han, Bing, Dalal, Siddhartha R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.02.2012; Elsevier
• Series: Computational Statistics & Data Analysis
• Subjects: Adjustment; Bernstein-type polynomials; Data processing; Decreasing density; Density; Density estimator; Density estimator Decreasing density Bernstein-type polynomials NPMLE; Estimates; Estimators; Exact sciences and technology; Gene expression; General topics; Mathematics; Multivariate analysis; Nonparametric inference; NPMLE; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Probability and statistics; Sciences and techniques of general use; Smoothness; Statistics; Density estimator; Bernstein-type polynomials; NPMLE; Decreasing density; Shape function; Density estimation; Data analysis; Non parametric estimation; Statistical estimation; Implementation; Bernstein polynomial; Hypothesis test; Statistical computation; Maximum likelihood; P value; Step function
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/j.csda.2011.08.010

The nonparametric maximum likelihood estimator (NPMLE) is a popular approach to estimating decreasing densities, i.e., f ( s ) ≥ f ( t ) , s ≤ t . A less ideal feature of NPMLE is its step-function form. In this paper, we propose two nonparametric density estimators based on the Bernstein-type polynomials of the NPMLE. The proposed estimators have relatively simple forms and easy implementation. They have satisfactory smoothness as well as estimation efficiency. Numerical examples demonstrate the superior performance of the proposed estimators compared to existing methods. Decreasing densities have been applied in simultaneous inference to estimate the proportion of true null hypotheses and the local false discovery rate. We applied the proposed estimators to conduct simultaneous tests for a gene expression data set.